McMullen mentions that Newton's method applied to q(x)=p(x)/r(x)=(x^3+a*x+b)/(3*a*x^2+9*b*x-a^2) is a more reliable iteration for solving the cubic p(x)=x^3+a*x+b than Newton's method applied to p(x)=x^3+a*x+b itself. "Braiding of the attractor and the failure of iterative algorithms", Invent. math. 91, 259-272 (1988). "Families of rational maps and iterative root-finding algorithms", Annals of Math., 125 (1987), 467-493. (Both papers are available for free online.) However, McMullen doesn't discuss where his algorithm came from, except the following quote in [Families...]: "The success of this algorithm can be understood by checking that the _points of inflection_ of q(x) coincide with the zeros of p(x). Thus Newton's method certainly converges if we take one of the points of inflection of q as our initial guess." I assume that the rationale for providing the denominator r(x) in q(x) is to avoid the well-known cycling discussed in good high school algebra classes. But McMullen doesn't provide any additional detail. I noticed that McMullen's approach wasn't discussed in the Wikipedia entry for cubics. Does anyone here have any thoughts on how to make McMullen's method more obvious?